Impact of Macroeconomic Factors on Apple Stock Price

There are many great technology sector companies to choose from, but one of the best is undoubtedly Apple Inc. (AAPL). Apple has consistently ranked among the world’s most valuable companies, with a market capitalization of over $2 trillion as of 2021.

Apple’s success can be attributed to its innovative products, such as the iPhone, iPad, and MacBook, as well as its ecosystem of software and services, including the App Store, iCloud, and Apple Music. The company also has a strong brand reputation and a loyal customer base.

Apple’s financial performance has been impressive, with steady revenue growth and strong profits. The company has consistently returned value to its shareholders through share buybacks and dividend payments.

By using an ARIMAX+ARCH/GARCH model to analyze the stock price behavior of Apple, we can gain insights into how macroeconomic factors impact the company’s performance. For example, an increase in GDP growth rate or a decrease in unemployment rate may lead to increased consumer spending and a rise in Apple’s stock price. Conversely, an increase in inflation or interest rates may lead to a decrease in consumer spending and a decline in Apple’s stock price.

Time series Plot

Apple

Code

# get data
options("getSymbols.warning4.0"=FALSE)
options("getSymbols.yahoo.warning"=FALSE)


data.info = getSymbols("AAPL",src='yahoo', from = '2010-01-01',to = "2023-03-01",auto.assign = FALSE)
data = getSymbols("AAPL",src='yahoo', from = '2010-01-01',to = "2023-03-01")
df <- data.frame(Date=index(AAPL),coredata(AAPL))

# create Bollinger Bands
bbands <- BBands(AAPL[,c("AAPL.High","AAPL.Low","AAPL.Close")])

# join and subset data
df <- subset(cbind(df, data.frame(bbands[,1:3])), Date >= "2010-01-01")


# colors column for increasing and decreasing
for (i in 1:length(df[,1])) {
  if (df$AAPL.Close[i] >= df$AAPL.Open[i]) {
      df$direction[i] = 'Increasing'
  } else {
      df$direction[i] = 'Decreasing'
  }
}

i <- list(line = list(color = '#922B21'))
d <- list(line = list(color = '#7F7F7F'))

# plot candlestick chart

fig <- df %>% plot_ly(x = ~Date, type="candlestick",
          open = ~AAPL.Open, close = ~AAPL.Close,
          high = ~AAPL.High, low = ~AAPL.Low, name = "AAPL",
          increasing = i, decreasing = d) 
fig <- fig %>% add_lines(x = ~Date, y = ~up , name = "B Bands",
            line = list(color = '#ccc', width = 0.5),
            legendgroup = "Bollinger Bands",
            hoverinfo = "none", inherit = F) 
fig <- fig %>% add_lines(x = ~Date, y = ~dn, name = "B Bands",
            line = list(color = '#ccc', width = 0.5),
            legendgroup = "Bollinger Bands", inherit = F,
            showlegend = FALSE, hoverinfo = "none") 
fig <- fig %>% add_lines(x = ~Date, y = ~mavg, name = "Mv Avg",
            line = list(color = '#C052B3', width = 0.5),
            hoverinfo = "none", inherit = F) 
fig <- fig %>% layout(yaxis = list(title = "Price"))

# plot volume bar chart
fig2 <- df 
fig2 <- fig2 %>% plot_ly(x=~Date, y=~AAPL.Volume, type='bar', name = "AAPL Volume",
          color = ~direction, colors = c('#922B21','#7F7F7F')) 
fig2 <- fig2 %>% layout(yaxis = list(title = "Volume"))

# create rangeselector buttons
rs <- list(visible = TRUE, x = 0.5, y = -0.055,
           xanchor = 'center', yref = 'paper',
           font = list(size = 9),
           buttons = list(
             list(count=1,
                  label='RESET',
                  step='all'),
             list(count=3,
                  label='3 YR',
                  step='year',
                  stepmode='backward'),
             list(count=1,
                  label='1 YR',
                  step='year',
                  stepmode='backward'),
             list(count=1,
                  label='1 MO',
                  step='month',
                  stepmode='backward')
           ))

# subplot with shared x axis
fig <- subplot(fig, fig2, heights = c(0.7,0.2), nrows=2,
             shareX = TRUE, titleY = TRUE)
fig <- fig %>% layout(title = paste("Apple Stock Price: January 2010 - March 2023"),
         xaxis = list(rangeselector = rs),
         legend = list(orientation = 'h', x = 0.5, y = 1,
                       xanchor = 'center', yref = 'paper',
                       font = list(size = 10),
                       bgcolor = 'transparent'))

fig

Differentitaed Chart Series

Code

log(data.info$`AAPL.Adjusted`) %>% diff() %>% chartSeries(theme=chartTheme('white'),up.col='#922B21')

GDP Growth

Code

#import the data
gdp <- read.csv("DATA/RAW DATA/gdp-growth.csv")

#change date format
gdp$Date <- as.Date(gdp$DATE , "%m/%d/%Y")

#drop DATE column
gdp <- subset(gdp, select = -c(1))

#export the cleaned data
gdp_clean <- gdp
write.csv(gdp_clean, "DATA/CLEANED DATA/gdp_clean_data.csv", row.names=FALSE)

#plot gdp growth rate 
fig <- plot_ly(gdp, x = ~Date, y = ~value, type = 'scatter', mode = 'lines',line = list(color = 'rgb(240, 128, 128)'))
fig <- fig %>% layout(title = "U.S GPD Growth Rate: 2010 - 2022",xaxis = list(title = "Time"),yaxis = list(title ="GDP Growth Rate"))
fig

Inflation

Code

#import the data
inflation_rate <- read.csv("DATA/RAW DATA/inflation-rate.csv")

#cleaning the data
#remove unwanted columns
inflation_rate_clean <- subset(inflation_rate, select = -c(1,HALF1,HALF2))

#convert the data to time series data
inflation_data_ts <- ts(as.vector(t(as.matrix(inflation_rate_clean))), start=c(2010,1), end=c(2023,2), frequency=12)

#export the data
write.csv(inflation_rate_clean, "DATA/CLEANED DATA/inflation_rate_clean_data.csv", row.names=FALSE)


#plot inflation rate 
fig <- autoplot(inflation_data_ts, ylab = "Inflation Rate", color="#FFA07A")+ggtitle("U.S Inflation Rate: January 2010 - February 2023")+theme_bw()
ggplotly(fig)

Interest

Code

#import the data
interest_data <- read.csv("DATA/RAW DATA/interest-rate.csv")

#change date format
interest_data$Date <- as.Date(interest_data$Date , "%m/%d/%Y")

#export the cleaned data
interest_clean_data <- interest_data
write.csv(interest_clean_data, "DATA/CLEANED DATA/interest_rate_clean_data.csv", row.names=FALSE)

#plot interest rate 
fig <- plot_ly(interest_data, x = ~Date, y = ~value, type = 'scatter', mode = 'lines',line = list(color='rgb(219, 112, 147)'))
fig <- fig %>% layout(title = "U.S Interest Rate: January 2010 - March 2023",xaxis = list(title = "Time"),yaxis = list(title ="Interest Rate"))
fig

Unemployment

Code

#import the data
unemployment_rate <- read.csv("DATA/RAW DATA/unemployment-rate.csv")

#change date format
unemployment_rate$Date <- as.Date(unemployment_rate$Date , "%m/%d/%Y")

# export the data
write.csv(unemployment_rate, "DATA/CLEANED DATA/unemployment_rate_clean_data.csv", row.names=FALSE)

#plot unemployment rate 
fig <- plot_ly(unemployment_rate, x = ~Date, y = ~Value, type = 'scatter', mode = 'lines',line = list(color = 'rgb(189, 183, 107)'))
fig <- fig %>% layout(title = "U.S Unemployment Rate: January 2010 - March 2023",xaxis = list(title = "Time"),yaxis = list(title ="Unemployment Rate"))
fig

Apple Inc.’s stock price has had a tumultuous journey from 2010 to 2023. In 2010, the company was just beginning to recover from the impact of the 2008 financial crisis, and its stock price was around $30 per share. However, the introduction of new products such as the iPad and iPhone in the following years propelled the company’s stock price to new heights, reaching over $700 per share in 2012.

From 2013 to 2016, Apple’s stock price continued to fluctuate, experiencing highs and lows due to factors such as changes in consumer demand and increased competition in the technology sector. The company’s stock price began to recover in 2017, as investors became more optimistic about the potential of Apple’s new products such as the iPhone X and the Apple Watch.

However, the outbreak of the COVID-19 pandemic in 2020 caused a brief dip in Apple’s stock price, as investors were uncertain about the impact of the pandemic on the company’s operations and financial performance. Nevertheless, Apple’s strong position in the technology sector and its ability to adapt to changing market conditions helped it to quickly rebound and continue its growth trend throughout 2020 and into early 2021.

Since mid 2020, Apple’s stock price has experienced some volatility, likely due to a combination of factors such as global economic uncertainty and fluctuations in consumer demand for the company’s products.

As discussed before, the macroeconomic factors of GDP growth, inflation, interest rates, and unemployment rate are closely interrelated and play a crucial role in the overall health and stability of an economy. From 2010 to 2023, the global economy experienced a mix of ups and downs, with periods of strong GDP growth followed by slowdowns and recessions.

The second plot shows the first difference of the logarithm of the adjusted Apple stock price. Taking the first difference removes any long-term trends and transforms the time series into a stationary process. From the plot, we can observe that the first difference of the logarithm of the Apple stock price appears to be stationary, as the mean and variance are roughly constant over time.

Enodogenous and Exogenous Variables

Plot

Code

numeric_data <- c("AAPL.Adjusted","gdp", "interest", "inflation", "unemployment")
numeric_data <- final[, numeric_data]
normalized_data_numeric <- scale(numeric_data)
normalized_data <- ts(normalized_data_numeric, start = c(2010, 1), end = c(2021,10),frequency = 4)
ts_plot(normalized_data,
        title = "Normalized Time Series Data for AAPL Stock and Macroeconomic Variables",
        Ytitle = "Normalized Values",
        Xtitle = "Year")

Correlation Heatmap

Code

# Get upper triangle of the correlation matrix
get_upper_tri <- function(cormat){
    cormat[lower.tri(cormat)]<- NA
    return(cormat)
}
cormat <- round(cor(normalized_data_numeric),2)

upper_tri <- get_upper_tri(cormat)

melted_cormat <- melt(upper_tri, na.rm = TRUE)
# Create a ggheatmap
ggheatmap <- ggplot(melted_cormat, aes(Var2, Var1, fill = value))+
 geom_tile(color = "white")+
 scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
   midpoint = 0, limit = c(-1,1), space = "Lab", 
    name="Pearson\nCorrelation") +
  theme_minimal()+ # minimal theme
 theme(axis.text.x = element_text(angle = 45, vjust = 1, 
    size = 12, hjust = 1))+
 coord_fixed()

ggheatmap + 
geom_text(aes(Var2, Var1, label = value), color = "black", size = 4) +
theme(
  axis.title.x = element_blank(),
  axis.title.y = element_blank(),
  panel.grid.major = element_blank(),
  panel.border = element_blank(),
  panel.background = element_blank(),
  axis.ticks = element_blank(),
  legend.justification = c(1, 0),
  legend.position = c(0.6, 0.7),
  legend.direction = "horizontal")+
  guides(fill = guide_colorbar(barwidth = 7, barheight = 1,
                title.position = "top", title.hjust = 0.5))

CCF GDP

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AAPL.Adjusted")], normalized_data[, c("gdp")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AAPL Stock Price and GDP Growth Rate ",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 5.344796

CCF Interest

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AAPL.Adjusted")], normalized_data[, c("interest")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AAPL Stock Price and Interest Rate",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 12.1218

CCF Inflation

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AAPL.Adjusted")], normalized_data[, c("inflation")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AAPL Stock Price and Inflation Rate",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 14.80688

CCF Unemployment

Code

par(mfrow=c(1,1))
ccf_result <- ccf(normalized_data[, c("AAPL.Adjusted")], normalized_data[, c("unemployment")], 
    lag.max = 300,
    main = "Cros-Correlation Plot for AAPL Stock Priceand Unemployment Rate",
    ylab = "CCF")

Code

cat("The sum of cross correlation function is", sum(abs(ccf_result$acf)))

The sum of cross correlation function is 18.29428

The Normalized Time Series Data for Stock Price and Macroeconomic Variables plot shows the same variables as the first plot but has been normalized to a common range of 0 to 1 using the scale() function in R, which standardizes the variables to have a mean of 0 and a standard deviation of 1. The heatmap analysis of the normalized data reveals that inflation and unemployment rate exhibit strong positive correlations with the stock price indices, indicating that these variables may significantly influence stock price movements. On the other hand, weaker correlations were observed between the stock price indices and GDP and interest rates, suggesting that these variables may have less impact on stock price fluctuations. The cross-correlation feature plots confirm these findings, indicating that inflation and unemployment rate are more suitable feature variables for the ARIMAX model when predicting Apple movements.

Final Exogenous variables: Macroeconomic indicators: Inflation rate and unemployment rate.

Enodogenous and Exogenous Variables Plot

Plot

Code

final_data <- final %>%dplyr::select( Date,AAPL.Adjusted, inflation,unemployment)
numeric_data <- c("AAPL.Adjusted", "inflation","unemployment")
numeric_data <- final_data[, numeric_data]
normalized_data_numeric <- scale(numeric_data)
normalized_numeric_df <- data.frame(normalized_data_numeric)
normalized_data_ts <- ts(normalized_data_numeric, start = c(2010, 1), frequency = 4)

autoplot(normalized_data_ts, facets=TRUE) +
  xlab("Year") + ylab("") +
  ggtitle("Apple Stock Price, Inflation Rate and Unemployment Rate in USA 2010-2023")

Check the stationarity

Code

# Convert your multivariate time series data to a matrix
final_data_ts_multivariate <- as.matrix(normalized_data_ts)

# Check for stationarity using Phillips-Perron test
phillips_perron_test <- ur.pp(final_data_ts_multivariate)  
summary(phillips_perron_test)

################################## 
# Phillips-Perron Unit Root Test # 
################################## 

Test regression with intercept 


Call:
lm(formula = y ~ y.l1)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.2155 -0.1522 -0.0540  0.0799  4.4890 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.000148   0.042080  -0.004    0.997    
y.l1         0.851949   0.042506  20.043   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5239 on 153 degrees of freedom
Multiple R-squared:  0.7242,    Adjusted R-squared:  0.7224 
F-statistic: 401.7 on 1 and 153 DF,  p-value: < 2.2e-16


Value of test-statistic, type: Z-alpha  is: -24.89 

         aux. Z statistics
Z-tau-mu           -0.0018

The results of the Phillips-Perron unit root test indicate strong evidence against the null hypothesis of a unit root, as the p-value for the coefficient of the lagged variable is less than the significance level of 0.05. This suggests that the variable y, which is being tested for stationarity, is likely stationary. Furthermore, the test statistic Z-tau-mu is -0.0018, which is smaller than the critical value of Z-alpha (-24.89), providing further evidence of stationarity.

To determine whether the linear model requires an ARCH model, an ARCH test is conducted. The ACF and PACF plots are also used to identify suitable model values.

Model Fitting

Plot

Code

normalized_numeric_df$AAPL.Adjusted<-ts(normalized_numeric_df$AAPL.Adjusted,star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
normalized_numeric_df$inflation<-ts(normalized_numeric_df$inflation,star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
normalized_numeric_df$unemployment<-ts(normalized_numeric_df$unemployment,star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)

fit <- lm(AAPL.Adjusted ~ inflation+unemployment, data=normalized_numeric_df)
fit.res<-ts(residuals(fit),star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
############## Then look at the residuals ############
returns <- fit.res  %>% diff()
autoplot(returns)+ggtitle("Linear Model Returns")

ARCH Test

Code

byd.archTest <- ArchTest(fit.res, lags = 1, demean = TRUE)
byd.archTest

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  fit.res
Chi-squared = 24.051, df = 1, p-value = 9.382e-07

ACF Plot

Code

ggAcf(returns) +ggtitle("ACF for returns")

PACF Plot

Code

ggPacf(returns) +ggtitle("PACF for returns")

The ARCH LM-test was conducted with the null hypothesis of no ARCH effects. The test resulted in a chi-squared value of 24.051 with one degree of freedom, and a very low p-value of 9.383e-07. This suggests strong evidence against the null hypothesis, indicating the presence of ARCH effects in the data.

Based on the ACF and PACF plots, it appears that there is some significant autocorrelation and partial autocorrelation at multiple lags, which suggests that an ARIMA model may not be sufficient to capture the time series behavior. Additionally, the values for p and q appear to be relatively high, with p = 0 and q = 0 being suggested by the plots.

ARIMAX Model

Auto Arima Model

Code

xreg <- cbind(Inflation = normalized_data_ts[, "inflation"],
              Unemployment = normalized_data_ts[, "unemployment"])
fit.auto <- auto.arima(normalized_data_ts[, "AAPL.Adjusted"], xreg = xreg)
summary(fit.auto)

Series: normalized_data_ts[, "AAPL.Adjusted"] 
Regression with ARIMA(0,1,0) errors 

Coefficients:
      Inflation  Unemployment
         0.1639       -0.1104
s.e.     0.0896        0.0407

sigma^2 = 0.04248:  log likelihood = 9.2
AIC=-12.4   AICc=-11.89   BIC=-6.61

Training set error measures:
                     ME      RMSE       MAE       MPE    MAPE      MASE
Training set 0.03812368 0.2000733 0.1274114 0.7800981 40.0008 0.4385178
                  ACF1
Training set 0.1228277

Auto Arima Residuals

Code

checkresiduals(fit.auto)

    Ljung-Box test

data:  Residuals from Regression with ARIMA(0,1,0) errors
Q* = 4.5496, df = 8, p-value = 0.8044

Model df: 0.   Total lags used: 8

Based on the results of the auto.arima function, the suggested best model is ARIMA(0,1,0) which is the same as the manual choosen arima model. So, we can then proceed to choose the best GARCH model using ARIMA(0,1,0) as the base model.

We can then proceed to choose the best GARCH model using ARIMA(0,1,0) as the base model.

Squared Residuals

Plot

Code

fit <- lm(AAPL.Adjusted ~ inflation+unemployment, data=normalized_numeric_df)
fit.res<-ts(residuals(fit),star=decimal_date(as.Date("2010-01-01",format = "%Y-%m-%d")),frequency = 4)
fit <- Arima(fit.res,order=c(0,1,0))
res=fit$res
plot(res^2,main='Squared Residuals')

ACF Plot

Code

acf(res^2,24, main = "ACF Residuals Square")

PACF Plot

Code

pacf(res^2,24, main = "PACF Residuals Square")

ACF Absolute

Code

acf(abs(res), main = "ACF of Absolute Residuals")

Code

pacf(abs(res), main = "PACF of Absolute Residuals")

ARCH Model

Code

summary(garchFit(~garch(1,0),res, trace=F))

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~garch(1, 0), data = res, trace = F) 

Mean and Variance Equation:
 data ~ garch(1, 0)
<environment: 0x13a282308>
 [data = res]

Conditional Distribution:
 norm 

Coefficient(s):
       mu      omega     alpha1  
0.0090525  0.0375268  0.6903879  

Std. Errors:
 based on Hessian 

Error Analysis:
        Estimate  Std. Error  t value Pr(>|t|)   
mu      0.009052    0.033063    0.274  0.78424   
omega   0.037527    0.012149    3.089  0.00201 **
alpha1  0.690388    0.365975    1.886  0.05924 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 -3.605681    normalized:  -0.06934001 

Description:
 Sat Jan  6 19:59:31 2024 by user:  


Standardised Residuals Tests:
                                Statistic p-Value  
 Jarque-Bera Test   R    Chi^2  2.85827   0.239516 
 Shapiro-Wilk Test  R    W      0.9672877 0.1618382
 Ljung-Box Test     R    Q(10)  7.91217   0.6374158
 Ljung-Box Test     R    Q(15)  8.892125  0.8830899
 Ljung-Box Test     R    Q(20)  11.42691  0.9343916
 Ljung-Box Test     R^2  Q(10)  12.0452   0.2820429
 Ljung-Box Test     R^2  Q(15)  12.59754  0.6333527
 Ljung-Box Test     R^2  Q(20)  14.78188  0.7887492
 LM Arch Test       R    TR^2   16.50618  0.1691362

Information Criterion Statistics:
      AIC       BIC       SIC      HQIC 
0.2540646 0.3666364 0.2478793 0.2972220 

From the squared residuals of the best ARIMA model, it can be observed that the ACF plot and PACF plot indicate that the residuals are not autocorrelated and are white noise, indicating a good fit of the model. Based on the squared residuals of the best ARIMA model, we can see that the ACF and PACF plots indicate that most of the values lie between the blue lines. Additionally, the p-value is 1 and q-value is 0. This suggests that the model has a good fit and that there is no significant autocorrelation or partial autocorrelation in the residuals.

The best models are ARIMA(0,1,0) and ARCH(1)

Best Model

ARIMA Model

Code

#fiting an ARIMA model to the Inflation variable
inflation_fit<-auto.arima(normalized_numeric_df$inflation) 
finflation<-forecast(inflation_fit)

#fitting an ARIMA model to the Unemployment variable
unemployment_fit<-auto.arima(normalized_numeric_df$unemployment) 
funemployment<-forecast(unemployment_fit)

# best model fit for forcasting
xreg <- cbind(Inflation = normalized_data_ts[, "inflation"],
              Unemployment = normalized_data_ts[, "unemployment"])



summary(arima.fit<-Arima(normalized_data_ts[, "AAPL.Adjusted"],order=c(0,1,0),xreg=xreg),include.drift = TRUE)

Series: normalized_data_ts[, "AAPL.Adjusted"] 
Regression with ARIMA(0,1,0) errors 

Coefficients:
      Inflation  Unemployment
         0.1639       -0.1104
s.e.     0.0896        0.0407

sigma^2 = 0.04248:  log likelihood = 9.2
AIC=-12.4   AICc=-11.89   BIC=-6.61

Training set error measures:
                     ME      RMSE       MAE       MPE    MAPE      MASE
Training set 0.03812368 0.2000733 0.1274114 0.7800981 40.0008 0.4385178
                  ACF1
Training set 0.1228277

GARCH Model

Code

summary(final.fit <- garchFit(~garch(1,0), res,trace = F))

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~garch(1, 0), data = res, trace = F) 

Mean and Variance Equation:
 data ~ garch(1, 0)
<environment: 0x13e4ee348>
 [data = res]

Conditional Distribution:
 norm 

Coefficient(s):
       mu      omega     alpha1  
0.0090525  0.0375268  0.6903879  

Std. Errors:
 based on Hessian 

Error Analysis:
        Estimate  Std. Error  t value Pr(>|t|)   
mu      0.009052    0.033063    0.274  0.78424   
omega   0.037527    0.012149    3.089  0.00201 **
alpha1  0.690388    0.365975    1.886  0.05924 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 -3.605681    normalized:  -0.06934001 

Description:
 Sat Jan  6 19:59:31 2024 by user:  


Standardised Residuals Tests:
                                Statistic p-Value  
 Jarque-Bera Test   R    Chi^2  2.85827   0.239516 
 Shapiro-Wilk Test  R    W      0.9672877 0.1618382
 Ljung-Box Test     R    Q(10)  7.91217   0.6374158
 Ljung-Box Test     R    Q(15)  8.892125  0.8830899
 Ljung-Box Test     R    Q(20)  11.42691  0.9343916
 Ljung-Box Test     R^2  Q(10)  12.0452   0.2820429
 Ljung-Box Test     R^2  Q(15)  12.59754  0.6333527
 Ljung-Box Test     R^2  Q(20)  14.78188  0.7887492
 LM Arch Test       R    TR^2   16.50618  0.1691362

Information Criterion Statistics:
      AIC       BIC       SIC      HQIC 
0.2540646 0.3666364 0.2478793 0.2972220 

Volatility

Code

ht <- final.fit@h.t #a numeric vector with the conditional variances (h.t = sigma.t^delta)

#############################
data=data.frame(final)
data$Date<-as.Date(data$Date,"%Y-%m-%d")


data2= data.frame(ht,data$Date)
ggplot(data2, aes(y = ht, x = data.Date)) + geom_line(col = '#922B21') + ylab('Conditional Variance') + xlab('Date')

From the ARIMA(0,1,0), we see that the training set error measures also suggest a good fit, with low mean absolute error, root mean squared error, and autocorrelation of the residuals. GATCH(1,0) model model is used to estimate the volatility of the standardized residuals of the previous regression model. The model includes a mean equation that estimates the mean of the residuals and a variance equation that models the conditional variance of the residuals. The coefficients of the mean equation suggest that the mean of the residuals is close to zero. The variance equation coefficients suggest that the conditional variance of the residuals is dependent on the past conditional variances and the past squared standardized residuals. The model’s AIC, BIC, SIC, and HQIC values are all relatively low, indicating a good fit of the model. The standardized residuals tests indicate that the residuals are approximately normally distributed and that there is no significant autocorrelation in the residuals.

The volatility of the model seems high in 2020 but has decreased gradually in the past few months. This could indicate that the asset’s price was experiencing a lot of fluctuations in 2020, but the market has stabilized recently.

Model Diagnostics

Residuals

Code

fit2<-garch(res,order=c(1,0),trace=F)
checkresiduals(fit2) 

Warning in modeldf.default(object): Could not find appropriate degrees of
freedom for this model.

QQ Plot

Code

qqnorm(fit2$residuals, pch = 1)
qqline(fit2$residuals, col = "blue", lwd = 2)

Box Test

Code

Box.test (fit2$residuals, type = "Ljung")

    Box-Ljung test

data:  fit2$residuals
X-squared = 2.6613, df = 1, p-value = 0.1028

The ACF plot of the residuals shows all the values between the blue lines, which indicates that the residuals are not significantly autocorrelated. The range of values for the residual plot between -2 and 2 is considered acceptable. Additionally, the QQ plot of the residuals shows a linear plot on the line, which is another good indication that the residuals are normally distributed. The QQ plot is a valuable tool to assess if the residuals follow a normal distribution, and in this case, the plot suggests that the residuals do indeed follow a normal distribution.

The Box-Ljung test, a p-value of 0.1028 indicates that the model’s residuals are not significantly autocorrelated, meaning that the model has captured most of the information in the data. This result is good because it suggests that the model is a good fit for the data and has accounted for most of the underlying patterns in the data. Therefore, we can rely on the model’s predictions and use them to make informed decisions.

Forecast

Code

predict(final.fit, n.ahead = 5, plot=TRUE)

  meanForecast meanError standardDeviation lowerInterval upperInterval
1   0.00905246 0.2478172         0.2478172    -0.4766603     0.4947653
2   0.00905246 0.2827116         0.2827116    -0.5450521     0.5631570
3   0.00905246 0.3044777         0.3044777    -0.5877128     0.6058177
4   0.00905246 0.3186383         0.3186383    -0.6154671     0.6335720
5   0.00905246 0.3280581         0.3280581    -0.6339296     0.6520345

The forecasted plot is based on the best model ARIMAX(0,1,0)+GARCH(1,0). This model takes into account the autoregressive and moving average components of the data, as well as the impact of exogenous variables on the time series. Additionally, the GARCH component of the model accounts for the volatility clustering in the data. Overall, this model is well-suited to make accurate predictions about future values of the time series.

Equation of the Model

The equation of the ARIMAX(0,1,0) model is:

\(Y(t) = Y(t-1) + \epsilon(t)\)

where \(Y(t)\) is the time series variable and \(\epsilon(t)\) is the error term.

The equation of the ARCH(1) model is:

\(\sigma_{t}^{2}=\alpha_{0}+\alpha_{1}\epsilon_{t-1}^{2}\)

where \(\sigma^2_t\) is the conditional variance at time \(t\), \(\alpha_0\) is a constant, \(\alpha_1\) and \(\beta_1\) are the parameters, and \(\epsilon_t\) is the error term.

The combined equation of the ARIMAX(1,1,1)+GARCH(1,1) model is:

\(Y(t) = Y(t-1) + \epsilon(t)\)

\(\sigma_{t}^{2}=\alpha_{0}+\alpha_{1}\epsilon_{t-1}^{2}\)